RLQ analysis

Prepare data

When preparing palm traits for analysis, I had to remove several variables that contained NAs for our palm species. Also, I removed the descriptive traits about the fruit, and the variable “FruitShape” because it has blank values.

Run RLQ analysis and plot data

To successfully run this, I had to remove Habitat type from our environmental variables. The problem might be the naming convention. Sarah, can you make three letter codes for these?

That’s unreadable, plotting as seperate.

Summary of RLQ analysis. How to interpret this?

## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = acpR.aravo, dudiL = afcL.aravo, dudiQ = acpQ.aravo, 
##     scannf = FALSE)
## 
## Total inertia: 0.657
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.300155 0.228303 0.083533 0.041707 0.001744 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 45.6877 34.7508 12.7148  6.3484  0.2654 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   45.69   80.44   93.15   99.50   99.77 
## 
## (Only 5 dimensions (out of 9) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 0.3001551 0.5478641 1.034229 1.481707 0.3575146
## 2 0.2283026 0.4778102 1.084627 2.004471 0.2197734
## 
## Inertia & coinertia R (acpR.aravo):
##     inertia      max     ratio
## 1  1.069629 1.755539 0.6092881
## 12 2.246045 3.085130 0.7280229
## 
## Inertia & coinertia Q (acpQ.aravo):
##     inertia      max     ratio
## 1  2.195457 5.095305 0.4308784
## 12 6.213361 7.950625 0.7814933
## 
## Correlation L (afcL.aravo):
##        corr       max     ratio
## 1 0.3575146 0.9327084 0.3833080
## 2 0.2197734 0.8335977 0.2636445

Fourth-corner analysis

From tutorial: “Fourth-corner analysis can be used to test the associations between individual traits and environmental variables. To obtain a test with a correct type I error, results of model 2 (permutation of sites, i.e. rows) and 4 (permutation of species, i.e. columns) should be combined.”

 nrepet <- 999
 four.comb.aravo <- fourthcorner(p_env[,-10], p_species,
     p_traits, modeltype = 6, p.adjust.method.G = "none",
     p.adjust.method.D = "none", nrepet = nrepet)

Plotting the data: “Blue cells correspond to negative significant relationships while red cells correspond to positive significant relationships (this can be modified using the argument col).”

I used the D2 option when plotting, but others exist: stat=“D2”: the association is measured between the quantitative variable and each category separately. A correlation coefficient is used to indicate the strength of the association between the given category and the small or large values of the quantitative variable. stat=“G”: the association between the quantitative variable and the whole categorical variable is measured by a global statistic (F). stat=“D”: the association is estimated between the quantitative variable and each category separately by a measure of the within-group homogeneity. The strength of the association is indicated by the dispersion of the values of the quantitative variable for a given category.

To replot the data for multiple comparisons: “Now, we adjust p-values for multiple comparisons (here we used the fdr method using the p.adjust.4thcorner function).”

Combine both approaches

“First, a multivariate test can be applied to evaluate the global significance of the traits-environment relationships. This test is based on the total inertia of the RLQ analysis”

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.aravo, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test       Obs    Std.Obs   Alter Pvalue
## 1 Model 2 0.6569709 -0.1346780 greater  0.507
## 2 Model 4 0.6569709 -0.9536772 greater  0.829

The total inertia of RLQ analysis is equal to the SRLQ multivariate statistic defined in Dray and Legendre (2008). This statistic is returned by the fourthcorner2 function:

Srlq <- fourthcorner2(p_env[,-10], p_species, p_traits,
     modeltype = 6, p.adjust.method.G = "fdr", nrepet = nrepet)
Srlq$trRLQ
## Monte-Carlo test
## Call: fourthcorner2(tabR = p_env[, -10], tabL = p_species, tabQ = p_traits, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: -8990.343 
## 
## Based on 999 replicates
## Simulated p-value: 0.822 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
## -9.206018e-01 -8.990088e+03  7.649906e-02

“Both approaches can be combined if RLQ scores are used to represent traits and environmental variables on a biplot. Then, significant associations revealed by the fourthcorner approach can be represented using segments (blue lines for negative associations, red lines for positive associations, see the argument col). Only traits and environmental variables that have at least one significant association are represented. Here, we apply this method using adjusted pvalues for multiple comparisons and a significant level α = 0.05.”

“Another approach is provided by the fourthcorner.rlq function and consists in testing directly the links between RLQ axes and traits (typetest=”Q.axes“) or environmental variables (typetest=”R.axes“).”

RLQ axes and traits

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.aravo, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat         Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.96134310  2.20116058      less
## 2                AxcR2 / Climb.0 Homog.  0.89121986 -0.53679743      less
## 3                AxcR1 / Climb.1 Homog.  0.03686797 -0.24848563      less
## 4                AxcR2 / Climb.1 Homog.  0.10785496  1.78642185      less
## 5                AxcR1 / Acaul.0 Homog.  1.00000000        <NA>      less
## 6                AxcR2 / Acaul.0 Homog.  1.00000000        <NA>      less
## 7                AxcR1 / Erect.0 Homog.  0.03686797 -0.24848563      less
## 8                AxcR2 / Erect.0 Homog.  0.10785496  1.78642185      less
## 9                AxcR1 / Erect.1 Homog.  0.96134310  2.20116058      less
## 10               AxcR2 / Erect.1 Homog.  0.89121986 -0.53679743      less
## 11               AxcR1 / StemS.0 Homog.  0.03698011 -0.82249690      less
## 12               AxcR2 / StemS.0 Homog.  0.10811460  1.69818917      less
## 13               AxcR1 / StemS.1 Homog.  0.48052421 -0.47715061      less
## 14               AxcR2 / StemS.1 Homog.  0.59926258  0.29260277      less
## 15               AxcR1 / StemS.2 Homog.  0.41908006  3.49481660      less
## 16               AxcR2 / StemS.2 Homog.  0.27599200  0.82078570      less
## 17               AxcR1 / StemA.0 Homog.  0.76595294  1.53669110      less
## 18               AxcR2 / StemA.0 Homog.  0.63295693 -1.43746503      less
## 19               AxcR1 / StemA.1 Homog.  0.20361638  0.26186245      less
## 20               AxcR2 / StemA.1 Homog.  0.32299828  0.83189141      less
## 21               AxcR1 / Leave.0 Homog.  0.72348658  2.13924819      less
## 22               AxcR2 / Leave.0 Homog.  0.52504761 -1.56755644      less
## 23               AxcR1 / Leave.1 Homog.  0.24149577  0.02111477      less
## 24               AxcR2 / Leave.1 Homog.  0.44244854  1.66729070      less
## 25       AxcR1 / MaxStemHeight_m      r  0.09149172  0.68994881 two-sided
## 26       AxcR2 / MaxStemHeight_m      r -0.14448147 -1.18998713 two-sided
## 27         AxcR1 / MaxStemDia_cm      r  0.08749172  0.67358979 two-sided
## 28         AxcR2 / MaxStemDia_cm      r -0.12934431 -1.06921687 two-sided
## 29          AxcR1 / Under.canopy Homog.  0.97761339  1.77069366      less
## 30          AxcR2 / Under.canopy Homog.  0.96901013  0.58548969      less
## 31     AxcR1 / Under.understorey Homog.  0.02065195 -0.45299627      less
## 32     AxcR2 / Under.understorey Homog.  0.03098774 -0.29627554      less
## 33 AxcR1 / AverageFruitLength_cm      r  0.24503760  1.89727497 two-sided
## 34 AxcR2 / AverageFruitLength_cm      r -0.12292261 -0.96520838 two-sided
## 35           AxcR1 / Fruit.large Homog.  0.42030052  2.10914139      less
## 36           AxcR2 / Fruit.large Homog.  0.51860246  2.78553943      less
## 37           AxcR1 / Fruit.small Homog.  0.55423824  2.73768704      less
## 38           AxcR2 / Fruit.small Homog.  0.46746559  1.25826119      less
## 39     AxcR1 / Consp.conspicuous Homog.  0.28302363 -1.35476992      less
## 40     AxcR2 / Consp.conspicuous Homog.  0.49950186  1.79562903      less
## 41         AxcR1 / Consp.cryptic Homog.  0.67206694  1.88678666      less
## 42         AxcR2 / Consp.cryptic Homog.  0.46811068 -0.15736019      less
##    Pvalue Pvalue.adj  
## 1   0.996          1  
## 2   0.261  0.8150625  
## 3    0.61  0.8150625  
## 4   0.946          1  
## 5       1          1  
## 6       1          1  
## 7    0.61  0.8150625  
## 8   0.946          1  
## 9   0.996          1  
## 10  0.261  0.8150625  
## 11  0.241  0.8150625  
## 12  0.942          1  
## 13  0.308  0.8150625  
## 14  0.584  0.8150625  
## 15  0.993          1  
## 16  0.833          1  
## 17  0.929          1  
## 18  0.099      0.693  
## 19  0.594  0.8150625  
## 20  0.811          1  
## 21  0.967          1  
## 22   0.07     0.6216  
## 23  0.538  0.8150625  
## 24  0.966          1  
## 25  0.527  0.8150625  
## 26  0.264  0.8150625  
## 27  0.524  0.8150625  
## 28  0.296  0.8150625  
## 29  0.973          1  
## 30  0.713          1  
## 31  0.504  0.8150625  
## 32  0.571  0.8150625  
## 33  0.046     0.6216  
## 34  0.366  0.8150625  
## 35  0.973          1  
## 36  0.993          1  
## 37  0.988          1  
## 38  0.911          1  
## 39  0.074     0.6216  
## 40  0.976          1  
## 41  0.944          1  
## 42  0.429  0.8150625  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

RLQ axes and environmental variables

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.aravo, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                          Test Stat          Obs     Std.Obs     Alter Pvalue
## 1        Canopy.Cover / AxcQ1    r -0.012420893 -0.08504096 two-sided  0.891
## 2  Understory.Density / AxcQ1    r -0.029892325 -0.28535590 two-sided  0.784
## 3         Leaf.Litter / AxcQ1    r  0.083803687  0.83347452 two-sided  0.432
## 4       Soil.Moisture / AxcQ1    r  0.122365689  1.10765079 two-sided  0.265
## 5                 Cec / AxcQ1    r  0.046971395  1.03900906 two-sided  0.333
## 6                 T50 / AxcQ1    r  0.009956983  0.17613818 two-sided  0.897
## 7                 T10 / AxcQ1    r  0.304644385  1.82411195 two-sided  0.079
## 8       Canopy.Height / AxcQ1    r -0.124070105 -1.03016040 two-sided  0.331
## 9           Elevation / AxcQ1    r  0.056259879  0.43232450 two-sided  0.687
## 10       Canopy.Cover / AxcQ2    r  0.014571724  0.12684783 two-sided  0.846
## 11 Understory.Density / AxcQ2    r -0.114617370 -1.15933364 two-sided   0.27
## 12        Leaf.Litter / AxcQ2    r  0.049781194  0.49259572 two-sided  0.636
## 13      Soil.Moisture / AxcQ2    r -0.068450849 -0.84961406 two-sided  0.419
## 14                Cec / AxcQ2    r  0.027732905  0.71951008 two-sided  0.499
## 15                T50 / AxcQ2    r  0.098808237  1.14865071 two-sided  0.219
## 16                T10 / AxcQ2    r -0.062682617 -0.36973774 two-sided  0.699
## 17      Canopy.Height / AxcQ2    r -0.090071839 -0.81666971 two-sided   0.48
## 18          Elevation / AxcQ2    r  0.117191890  0.95719518 two-sided  0.367
##           Pvalue.adj  
## 1              0.891  
## 2              0.882  
## 3  0.816545454545455  
## 4  0.777857142857143  
## 5  0.816545454545455  
## 6              0.897  
## 7              0.708  
## 8  0.816545454545455  
## 9  0.834352941176471  
## 10             0.891  
## 11 0.777857142857143  
## 12 0.834352941176471  
## 13 0.816545454545455  
## 14 0.816545454545455  
## 15 0.777857142857143  
## 16 0.834352941176471  
## 17 0.816545454545455  
## 18 0.816545454545455  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Results can be represented using a table with colors indicating significance :

Significance with axes can also be reported on the factorial map of RLQ analysis. Here, significant associations with the first axis are represented in blue, with the second axis in orange, with both axes in green (variables with no significant association are in black)

RUN 2 RLQ/4th corner: Splitting all lifestage and points with some forest cover

Run RLQ/ FQ analysis for all lifestage types

That’s unreadable, plotting as separate.

## [1] "RLQ for juveniles"

## [1] "RLQ for adults"

Summary of RLQ analysis. How to interpret this?

## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Rjuv, dudiL = Ljuv, dudiQ = Qjuv, scannf = FALSE)
## 
## Total inertia: 1.429
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.746537 0.500895 0.121014 0.054025 0.003262 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 52.2481 35.0563  8.4695  3.7811  0.2283 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   52.25   87.30   95.77   99.55   99.78 
## 
## (Only 5 dimensions (out of 9) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 0.7465371 0.8640238 1.170836 1.785317 0.4133466
## 2 0.5008948 0.7077392 1.244773 1.691688 0.3360957
## 
## Inertia & coinertia R (Rjuv):
##     inertia      max     ratio
## 1  1.370856 1.858611 0.7375703
## 12 2.920316 3.464436 0.8429412
## 
## Inertia & coinertia Q (Qjuv):
##     inertia      max     ratio
## 1  3.187357 5.160919 0.6175949
## 12 6.049164 7.993775 0.7567344
## 
## Correlation L (Ljuv):
##        corr       max     ratio
## 1 0.4133466 0.9448971 0.4374514
## 2 0.3360957 0.9087471 0.3698452
## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Radu, dudiL = Ladu, dudiQ = Qadu, scannf = FALSE)
## 
## Total inertia: 1.04
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.564979 0.338546 0.115397 0.015354 0.003838 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
##  54.314  32.546  11.094   1.476   0.369 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   54.31   86.86   97.95   99.43   99.80 
## 
## (Only 5 dimensions (out of 10) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 0.5649794 0.7516511 1.317944 1.702729 0.3349451
## 2 0.3385457 0.5818468 1.128801 2.171484 0.2373747
## 
## Inertia & coinertia R (Radu):
##     inertia      max     ratio
## 1  1.736977 2.095715 0.8288231
## 12 3.011170 3.680365 0.8181714
## 
## Inertia & coinertia Q (Qadu):
##     inertia      max     ratio
## 1  2.899287 4.937157 0.5872382
## 12 7.614631 7.890549 0.9650318
## 
## Correlation L (Ladu):
##        corr       max     ratio
## 1 0.3349451 1.0000000 0.3349451
## 2 0.2373747 0.9128287 0.2600430

Fourth-corner analysis

## [1] "FQ for juveniles"

## [1] "FQ for adults"

With adjusted p-values

## [1] "FQ for Juveniles"
## [1] "FQ for Adults"

Combine both approaches

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs    Std.Obs   Alter Pvalue
## 1 Model 2 1.442008  0.2686947 greater  0.367
## 2 Model 4 1.442008 -0.3683070 greater  0.599

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs    Std.Obs   Alter Pvalue
## 1 Model 2 1.035941 15.3419891 greater  0.001
## 2 Model 4 1.035941 -0.1279593 greater  0.503

The total inertia of RLQ analysis is equal to the SRLQ multivariate statistic defined in Dray and Legendre (2008). This statistic is returned by the fourthcorner2 function:

## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envFORCOVER, tabL = p_speciesJUV, tabQ = p_traits[, 
##     -2], modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 1.442008 
## 
## Based on 858 replicates
## Simulated p-value: 0.5518044 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
##  3.893864e-01 -1.853161e+03  2.268507e+07
## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envFORCOVER, tabL = p_speciesADU, tabQ = p_traits[, 
##     -2], modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 1.035941 
## 
## Based on 999 replicates
## Simulated p-value: 0.487 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
##  3.627491e-01 -1.798911e+03  2.462105e+07
## [1] "juvenile"

## [1] "adult"

“Another approach is provided by the fourthcorner.rlq function and consists in testing directly the links between RLQ axes and traits (typetest=”Q.axes“) or environmental variables (typetest=”R.axes“).”

RLQ axes and traits

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat          Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.943613890  1.38352846      less
## 2                AxcR2 / Climb.0 Homog.  0.909518009 -0.29764535      less
## 3                AxcR1 / Climb.1 Homog.  0.053447423 -0.05495846      less
## 4                AxcR2 / Climb.1 Homog.  0.084371434  0.16475841      less
## 5                AxcR1 / Erect.0 Homog.  0.053447423 -0.05495846      less
## 6                AxcR2 / Erect.0 Homog.  0.084371434  0.16475841      less
## 7                AxcR1 / Erect.1 Homog.  0.943613890  1.38352846      less
## 8                AxcR2 / Erect.1 Homog.  0.909518009 -0.29764535      less
## 9                AxcR1 / StemS.0 Homog.  0.053447423 -0.49833525      less
## 10               AxcR2 / StemS.0 Homog.  0.084371434  0.16475841      less
## 11               AxcR1 / StemS.1 Homog.  0.428025862 -0.73020926      less
## 12               AxcR2 / StemS.1 Homog.  0.521201486 -0.21515194      less
## 13               AxcR1 / StemS.2 Homog.  0.347338026  2.27271748      less
## 14               AxcR2 / StemS.2 Homog.  0.386202949  2.87861842      less
## 15               AxcR1 / StemA.0 Homog.  0.826344445  2.38518552      less
## 16               AxcR2 / StemA.0 Homog.  0.576845878 -1.70242052      less
## 17               AxcR1 / StemA.1 Homog.  0.170876549  0.03530479      less
## 18               AxcR2 / StemA.1 Homog.  0.292200392  0.78366931      less
## 19               AxcR1 / Leave.0 Homog.  0.768813593  2.58506552      less
## 20               AxcR2 / Leave.0 Homog.  0.492148977 -1.63730148      less
## 21               AxcR1 / Leave.1 Homog.  0.224844454 -0.02492680      less
## 22               AxcR2 / Leave.1 Homog.  0.424884198  1.62034436      less
## 23       AxcR1 / MaxStemHeight_m      r -0.208407212 -1.35478741 two-sided
## 24       AxcR2 / MaxStemHeight_m      r -0.078947746 -0.64599103 two-sided
## 25         AxcR1 / MaxStemDia_cm      r -0.209187895 -1.35875846 two-sided
## 26         AxcR2 / MaxStemDia_cm      r -0.054305894 -0.49393222 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.981347340  0.91793232      less
## 28          AxcR2 / Under.canopy Homog.  0.981120201  0.87789559      less
## 29     AxcR1 / Under.understorey Homog.  0.017227001 -0.55517715      less
## 30     AxcR2 / Under.understorey Homog.  0.018790061 -0.54651300      less
## 31 AxcR1 / AverageFruitLength_cm      r -0.412507031 -2.73863068 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r -0.007459463 -0.08050279 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.381757814  1.54708263      less
## 34           AxcR2 / Fruit.large Homog.  0.473864901  2.21056477      less
## 35           AxcR1 / Fruit.small Homog.  0.520672457  2.44145746      less
## 36           AxcR2 / Fruit.small Homog.  0.525158287  2.34585118      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.257598643 -1.27517886      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.462805149  1.52340090      less
## 39         AxcR1 / Consp.cryptic Homog.  0.731937157  2.46836745      less
## 40         AxcR2 / Consp.cryptic Homog.  0.452042297 -0.22024949      less
##               Pvalue        Pvalue.adj  
## 1              0.943             0.996  
## 2              0.249 0.845882352941176  
## 3  0.640229885057471 0.845882352941176  
## 4              0.608             0.996  
## 5  0.640229885057471 0.845882352941176  
## 6              0.608             0.996  
## 7              0.943             0.996  
## 8              0.249 0.845882352941176  
## 9  0.457661290322581 0.845882352941176  
## 10             0.608             0.996  
## 11             0.268 0.845882352941176  
## 12             0.432 0.845882352941176  
## 13             0.962             0.996  
## 14             0.984             0.996  
## 15             0.992             0.996  
## 16             0.057             0.528  
## 17             0.559 0.845882352941176  
## 18             0.785             0.996  
## 19             0.996             0.996  
## 20             0.066             0.528  
## 21             0.531 0.845882352941176  
## 22             0.961             0.996  
## 23             0.197 0.845882352941176  
## 24             0.563             0.996  
## 25             0.181 0.845882352941176  
## 26             0.664             0.996  
## 27             0.833             0.996  
## 28               0.8             0.996  
## 29 0.484814398200225 0.845882352941176  
## 30 0.565804274465692 0.845882352941176  
## 31             0.001              0.04 *
## 32             0.947             0.996  
## 33             0.934 0.957948717948718  
## 34             0.979             0.979  
## 35             0.981             0.996  
## 36             0.973             0.996  
## 37             0.118 0.786666666666667  
## 38             0.955             0.996  
## 39             0.995             0.996  
## 40              0.43 0.845882352941176  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat         Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.83426767 -1.24218503      less
## 2                AxcR2 / Climb.0 Homog.  0.95099413  0.87198368      less
## 3                AxcR1 / Climb.1 Homog.  0.07511623  1.20481129      less
## 4                AxcR2 / Climb.1 Homog.  0.04845745 -0.10705234      less
## 5                AxcR1 / Erect.0 Homog.  0.07511623  1.20481129      less
## 6                AxcR2 / Erect.0 Homog.  0.04845745 -0.10705234      less
## 7                AxcR1 / Erect.1 Homog.  0.83426767 -1.24218503      less
## 8                AxcR2 / Erect.1 Homog.  0.95099413  0.87198368      less
## 9                AxcR1 / StemS.0 Homog.  0.09973008  2.81986579      less
## 10               AxcR2 / StemS.0 Homog.  0.05482648 -0.54811255      less
## 11               AxcR1 / StemS.1 Homog.  0.64768119  0.62599905      less
## 12               AxcR2 / StemS.1 Homog.  0.71039950  0.83245462      less
## 13               AxcR1 / StemS.2 Homog.  0.15422535 -0.96229547      less
## 14               AxcR2 / StemS.2 Homog.  0.20727085 -0.54834780      less
## 15               AxcR1 / StemA.0 Homog.  0.63755477  1.50219243      less
## 16               AxcR2 / StemA.0 Homog.  0.62430580  0.69773950      less
## 17               AxcR1 / StemA.1 Homog.  0.33541957  1.41660183      less
## 18               AxcR2 / StemA.1 Homog.  0.36632872  1.46725454      less
## 19               AxcR1 / Leave.0 Homog.  0.48818798 -1.92147038      less
## 20               AxcR2 / Leave.0 Homog.  0.57383859  1.10112628      less
## 21               AxcR1 / Leave.1 Homog.  0.51160198  2.25099033      less
## 22               AxcR2 / Leave.1 Homog.  0.41483194  1.19671885      less
## 23       AxcR1 / MaxStemHeight_m      r -0.05356163 -0.29065001 two-sided
## 24       AxcR2 / MaxStemHeight_m      r  0.25025122  1.87017083 two-sided
## 25         AxcR1 / MaxStemDia_cm      r  0.07037235  0.50151065 two-sided
## 26         AxcR2 / MaxStemDia_cm      r  0.24386735  1.88078829 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.93589281  0.18237934      less
## 28          AxcR2 / Under.canopy Homog.  0.94840727  1.04665115      less
## 29     AxcR1 / Under.understorey Homog.  0.06357424  0.21559385      less
## 30     AxcR2 / Under.understorey Homog.  0.04964550 -0.04992555      less
## 31 AxcR1 / AverageFruitLength_cm      r  0.02508289  0.20088746 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r  0.15744770  1.18935170 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.45555129  2.44743065      less
## 34           AxcR2 / Fruit.large Homog.  0.55644465  2.86621912      less
## 35           AxcR1 / Fruit.small Homog.  0.53904796  3.44626257      less
## 36           AxcR2 / Fruit.small Homog.  0.40570765 -1.51656550      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.57936912  2.11601780      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.50416577  0.03958443      less
## 39         AxcR1 / Consp.cryptic Homog.  0.42033419 -0.41412270      less
## 40         AxcR2 / Consp.cryptic Homog.  0.47615401 -0.05341246      less
##               Pvalue        Pvalue.adj  
## 1               0.07               0.4  
## 2              0.817             0.999  
## 3               0.88             0.999  
## 4  0.650627615062761 0.968484848484849  
## 5               0.88             0.999  
## 6  0.650627615062761 0.968484848484849  
## 7               0.07               0.4  
## 8              0.817             0.999  
## 9              0.997             0.999  
## 10             0.402 0.861052631578947  
## 11             0.709 0.968484848484849  
## 12             0.804             0.999  
## 13               0.2 0.727272727272727  
## 14              0.35 0.861052631578947  
## 15             0.938             0.999  
## 16             0.763             0.999  
## 17             0.897  0.96972972972973  
## 18             0.877  0.96972972972973  
## 19             0.034               0.4  
## 20             0.867             0.999  
## 21             0.985             0.999  
## 22             0.846  0.96972972972973  
## 23             0.799 0.968484848484849  
## 24             0.049               0.4  
## 25             0.646 0.968484848484849  
## 26             0.052               0.4  
## 27             0.557 0.912307692307692  
## 28             0.844             0.999  
## 29 0.667731629392971 0.968484848484849  
## 30 0.667731629392971 0.968484848484849  
## 31             0.866  0.96972972972973  
## 32             0.261 0.861052631578947  
## 33             0.992             0.999  
## 34             0.999             0.999  
## 35             0.999             0.999  
## 36             0.063              0.21  
## 37             0.978             0.999  
## 38             0.498 0.948571428571429  
## 39             0.358 0.861052631578947  
## 40             0.491 0.948571428571429  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

RLQ axes and environmental variables

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                          Test   Stat          Obs     Std.Obs     Alter Pvalue        Pvalue.adj  
## 1        Canopy.Cover / AxcQ1      r -0.011478310 -0.09435422 two-sided   0.91 0.953333333333333  
## 2  Understory.Density / AxcQ1      r -0.043904212 -0.26121892 two-sided  0.804            0.9273  
## 3         Leaf.Litter / AxcQ1      r -0.005484529 -0.02763386 two-sided  0.981             0.981  
## 4       Soil.Moisture / AxcQ1      r -0.225084984 -1.63397166 two-sided   0.12             0.528  
## 5                 Cec / AxcQ1      r -0.023971630 -0.16454271 two-sided  0.725            0.9273  
## 6                 T50 / AxcQ1      r  0.044249272  0.59011481 two-sided  0.614 0.898333333333333  
## 7                 T10 / AxcQ1      r -0.325127892 -1.54553960 two-sided  0.129 0.520666666666667  
## 8       Canopy.Height / AxcQ1      r  0.092030583  0.62102573 two-sided  0.591 0.898333333333333  
## 9           Elevation / AxcQ1      r  0.070324854  0.42198134 two-sided   0.71 0.898333333333333  
## 10      Habit.Primary / AxcQ1 Homog.  0.331681420 -1.73563916      less  0.069            0.3795  
## 11    Habit.Secondary / AxcQ1 Homog.  0.599381420  1.47891320      less  0.912             0.953  
## 12       Canopy.Cover / AxcQ2      r -0.097263882 -1.26138728 two-sided  0.222             0.682  
## 13 Understory.Density / AxcQ2      r -0.200252450 -1.67024964 two-sided   0.09             0.495  
## 14        Leaf.Litter / AxcQ2      r  0.158914017  1.23240498 two-sided  0.251             0.682  
## 15      Soil.Moisture / AxcQ2      r -0.075420075 -0.66916283 two-sided   0.54 0.898333333333333  
## 16                Cec / AxcQ2      r  0.054906583  0.90105574 two-sided  0.394             0.796  
## 17                T50 / AxcQ2      r  0.158470974  1.47641432 two-sided   0.17 0.534285714285714  
## 18                T10 / AxcQ2      r  0.112190989  0.56034534 two-sided  0.591 0.898333333333333  
## 19      Canopy.Height / AxcQ2      r -0.123081690 -0.94914207 two-sided  0.398             0.796  
## 20          Elevation / AxcQ2      r  0.223865723  1.55322575 two-sided  0.142 0.520666666666667  
## 21      Habit.Primary / AxcQ2 Homog.  0.552321876  1.02953064      less  0.843            0.9273  
## 22    Habit.Secondary / AxcQ2 Homog.  0.447676330 -0.57625112      less  0.279             0.682  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                          Test   Stat          Obs    Std.Obs     Alter Pvalue        Pvalue.adj  
## 1        Canopy.Cover / AxcQ1      r -0.036559173 -0.8291895 two-sided  0.449             0.682  
## 2  Understory.Density / AxcQ1      r -0.191817215 -2.2067015 two-sided  0.045             0.396  
## 3         Leaf.Litter / AxcQ1      r  0.065828199  0.7223843 two-sided  0.527             0.682  
## 4       Soil.Moisture / AxcQ1      r  0.195438294  1.9825725 two-sided  0.073            0.4015  
## 5                 Cec / AxcQ1      r  0.049119486  1.0689861 two-sided  0.323 0.619666666666667  
## 6                 T50 / AxcQ1      r -0.069148216 -1.7669571 two-sided  0.054             0.396  
## 7                 T10 / AxcQ1      r -0.094793506 -0.9921533 two-sided  0.335 0.619666666666667  
## 8       Canopy.Height / AxcQ1      r  0.111040051  0.8764882 two-sided  0.333 0.619666666666667  
## 9           Elevation / AxcQ1      r -0.282626507 -2.0599475 two-sided  0.041             0.396  
## 10      Habit.Primary / AxcQ1 Homog.  0.629292241  4.4138312      less      1                 1  
## 11    Habit.Secondary / AxcQ1 Homog.  0.370594752  0.1728109      less  0.506             0.682  
## 12       Canopy.Cover / AxcQ2      r -0.050123436 -1.2979342 two-sided   0.21 0.619666666666667  
## 13 Understory.Density / AxcQ2      r -0.025867777 -0.3116514 two-sided  0.759 0.875809523809524  
## 14        Leaf.Litter / AxcQ2      r  0.117444694  1.3798924 two-sided  0.182 0.619666666666667  
## 15      Soil.Moisture / AxcQ2      r -0.077549281 -0.7294831 two-sided  0.492             0.682  
## 16                Cec / AxcQ2      r  0.004084052  0.1301069 two-sided  0.905            0.9955  
## 17                T50 / AxcQ2      r -0.009674346 -0.2513624 two-sided   0.82 0.949473684210526  
## 18                T10 / AxcQ2      r  0.124474996  1.1152124 two-sided  0.338 0.619666666666667  
## 19      Canopy.Height / AxcQ2      r  0.179515612  1.4784698 two-sided  0.149 0.619666666666667  
## 20          Elevation / AxcQ2      r  0.030718449  0.2160426 two-sided  0.833 0.875809523809524  
## 21      Habit.Primary / AxcQ2 Homog.  0.685658320  7.0650127      less      1                 1  
## 22    Habit.Secondary / AxcQ2 Homog.  0.313497605 -0.7161224      less  0.238 0.619666666666667  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Results can be represented using a table with colors indicating significance :

## [1] "juveniles"

## [1] "adults"

Significance with axes can also be reported on the factorial map of RLQ analysis. Here, significant associations with the first axis are represented in blue, with the second axis in orange, with both axes in green (variables with no significant association are in black)

## [1] "juveniles"

## [1] "adults"

RUN 3 RLQ/4th corner : Sarah data with endemism, distance to edge

First,Checking to see which environmental variables may be removed. Next, adding “endemism” as a trait and distance to edge as an environmental variable. Also, removing acualescence as a trait because it is 0 for all species

Run RLQ/ FQ analysis for all lifestage types with aforementioned additions and subtractions

That’s unreadable, plotting as separate.

## [1] "RLQ for juveniles"

## [1] "RLQ for adults"

Summary of RLQ analysis. How to interpret this?

## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Rjuv, dudiL = Ljuv, dudiQ = Qjuv, scannf = FALSE)
## 
## Total inertia: 1.863
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 1.026449 0.591536 0.170240 0.060074 0.009378 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 55.1060 31.7572  9.1395  3.2251  0.5035 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   55.11   86.86   96.00   99.23   99.73 
## 
## (Only 5 dimensions (out of 10) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 1.0264493 1.0131383 1.159306 1.951706 0.4477715
## 2 0.5915357 0.7691136 1.330778 1.730192 0.3340340
## 
## Inertia & coinertia R (Rjuv):
##     inertia      max     ratio
## 1  1.343990 2.065940 0.6505464
## 12 3.114959 3.663509 0.8502666
## 
## Inertia & coinertia Q (Qjuv):
##     inertia      max     ratio
## 1  3.809156 5.920973 0.6433329
## 12 6.802721 8.827788 0.7706030
## 
## Correlation L (Ljuv):
##        corr       max     ratio
## 1 0.4477715 0.9445514 0.4740573
## 2 0.3340340 0.9087441 0.3675776
## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Radu, dudiL = Ladu, dudiQ = Qadu, scannf = FALSE)
## 
## Total inertia: 1.209
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.645060 0.394890 0.143635 0.014464 0.008915 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 53.3419 32.6546 11.8776  1.1961  0.7372 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   53.34   86.00   97.87   99.07   99.81 
## 
## (Only 5 dimensions (out of 11) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 0.6450601 0.8031564 1.359344 1.800936 0.3280747
## 2 0.3948903 0.6284030 1.165037 2.240272 0.2407674
## 
## Inertia & coinertia R (Radu):
##     inertia      max     ratio
## 1  1.847815 2.118831 0.8720918
## 12 3.205126 3.967985 0.8077466
## 
## Inertia & coinertia Q (Qadu):
##     inertia      max     ratio
## 1  3.243369 5.609379 0.5782047
## 12 8.262189 8.631936 0.9571652
## 
## Correlation L (Ladu):
##        corr       max     ratio
## 1 0.3280747 1.0000000 0.3280747
## 2 0.2407674 0.9128018 0.2637675

Fourth-corner analysis

## [1] "FQ for juveniles"

## [1] "FQ for adults"

With adjusted p-values

## [1] "FQ for Juveniles"
## [1] "FQ for Adults"

Combine both approaches

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs    Std.Obs   Alter Pvalue
## 1 Model 2 1.862683  0.4309029 greater  0.312
## 2 Model 4 1.862683 -0.3240938 greater  0.580

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs      Std.Obs   Alter Pvalue
## 1 Model 2 1.209293 14.908845364 greater  0.001
## 2 Model 4 1.209293 -0.004556487 greater  0.465

The total inertia of RLQ analysis is equal to the SRLQ multivariate statistic defined in Dray and Legendre (2008). This statistic is returned by the fourthcorner2 function:

## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envFORCOVER1, tabL = p_speciesJUV, tabQ = p_traits1, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 1.862683 
## 
## Based on 883 replicates
## Simulated p-value: 0.4852941 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
##  3.648361e-01 -2.034443e+03  3.115231e+07
## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envFORCOVER1, tabL = p_speciesADU, tabQ = p_traits1, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 1.209293 
## 
## Based on 999 replicates
## Simulated p-value: 0.431 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
##  3.499361e-01 -1.878783e+03  2.886256e+07
## [1] "juvenile"

## [1] "adult"

“Another approach is provided by the fourthcorner.rlq function and consists in testing directly the links between RLQ axes and traits (typetest=”Q.axes“) or environmental variables (typetest=”R.axes“).”

RLQ axes and traits

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat          Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.950193251  1.78473803      less
## 2                AxcR2 / Climb.0 Homog.  0.898922998 -0.42095161      less
## 3                AxcR1 / Climb.1 Homog.  0.049695209 -0.11445033      less
## 4                AxcR2 / Climb.1 Homog.  0.089668424  0.42121411      less
## 5                AxcR1 / Erect.0 Homog.  0.049695209 -0.11445033      less
## 6                AxcR2 / Erect.0 Homog.  0.089668424  0.42121411      less
## 7                AxcR1 / Erect.1 Homog.  0.950193251  1.78473803      less
## 8                AxcR2 / Erect.1 Homog.  0.898922998 -0.42095161      less
## 9                AxcR1 / StemS.0 Homog.  0.049695209 -0.59385264      less
## 10               AxcR2 / StemS.0 Homog.  0.089668424  0.42121411      less
## 11               AxcR1 / StemS.1 Homog.  0.368448080 -1.06661457      less
## 12               AxcR2 / StemS.1 Homog.  0.525133339 -0.12897045      less
## 13               AxcR1 / StemS.2 Homog.  0.403487598  3.43678672      less
## 14               AxcR2 / StemS.2 Homog.  0.371167341  2.88335254      less
## 15               AxcR1 / StemA.0 Homog.  0.847080961  3.09997508      less
## 16               AxcR2 / StemA.0 Homog.  0.560193193 -1.86718901      less
## 17               AxcR1 / StemA.1 Homog.  0.141642553 -0.18172266      less
## 18               AxcR2 / StemA.1 Homog.  0.312050411  1.35097765      less
## 19               AxcR1 / Leave.0 Homog.  0.797333053  3.40669292      less
## 20               AxcR2 / Leave.0 Homog.  0.468235999 -1.84730628      less
## 21               AxcR1 / Leave.1 Homog.  0.194270324 -0.25491466      less
## 22               AxcR2 / Leave.1 Homog.  0.460207772  2.41514864      less
## 23       AxcR1 / MaxStemHeight_m      r -0.234799983 -1.34026471 two-sided
## 24       AxcR2 / MaxStemHeight_m      r -0.066945580 -0.51622786 two-sided
## 25         AxcR1 / MaxStemDia_cm      r -0.253003740 -1.42089696 two-sided
## 26         AxcR2 / MaxStemDia_cm      r -0.035305477 -0.29971927 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.979869766  0.66808379      less
## 28          AxcR2 / Under.canopy Homog.  0.981534581  0.95372648      less
## 29     AxcR1 / Under.understorey Homog.  0.018403668 -0.52356692      less
## 30     AxcR2 / Under.understorey Homog.  0.018394487 -0.52136764      less
## 31 AxcR1 / AverageFruitLength_cm      r -0.451443234 -2.54658724 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r -0.004210788 -0.03114628 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.325095192  1.27815147      less
## 34           AxcR2 / Fruit.large Homog.  0.476489677  2.26311991      less
## 35           AxcR1 / Fruit.small Homog.  0.550771380  3.01380670      less
## 36           AxcR2 / Fruit.small Homog.  0.523098288  2.51779907      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.220090824 -1.60012966      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.498734672  2.26617123      less
## 39         AxcR1 / Consp.cryptic Homog.  0.765977477  3.36043889      less
## 40         AxcR2 / Consp.cryptic Homog.  0.427568014 -0.34961875      less
## 41               AxcR1 / Endem.N Homog.  0.648448960  2.16671582      less
## 42               AxcR2 / Endem.N Homog.  0.671464239  2.50223484      less
## 43               AxcR1 / Endem.Y Homog.  0.248817870  0.63722025      less
## 44               AxcR2 / Endem.Y Homog.  0.310604637  1.00862386      less
##               Pvalue        Pvalue.adj  
## 1              0.988                 1  
## 2               0.32 0.914064516129032  
## 3  0.634593356242841 0.914064516129032  
## 4              0.707                 1  
## 5  0.634593356242841 0.914064516129032  
## 6              0.707                 1  
## 7              0.988                 1  
## 8               0.32 0.914064516129032  
## 9   0.38109756097561 0.914064516129032  
## 10             0.707                 1  
## 11             0.151            0.6908  
## 12             0.481 0.914064516129032  
## 13             0.997                 1  
## 14             0.985                 1  
## 15                 1                 1  
## 16              0.07 0.513333333333333  
## 17             0.493 0.914064516129032  
## 18             0.914                 1  
## 19                 1                 1  
## 20             0.038             0.418  
## 21             0.424 0.914064516129032  
## 22             0.996                 1  
## 23              0.19              0.76  
## 24             0.639                 1  
## 25             0.157            0.6908  
## 26             0.794                 1  
## 27             0.732                 1  
## 28             0.834                 1  
## 29 0.252595155709343 0.914064516129032  
## 30 0.565167243367935 0.914064516129032  
## 31             0.001             0.044 *
## 32             0.973                 1  
## 33             0.874            0.9614  
## 34             0.992             0.992  
## 35             0.996                 1  
## 36             0.983                 1  
## 37             0.053            0.4664  
## 38             0.995                 1  
## 39             0.999                 1  
## 40             0.365 0.914064516129032  
## 41             0.979                 1  
## 42             0.991                 1  
## 43             0.744 0.935314285714286  
## 44             0.831            0.9614  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat         Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog. 0.839889789 -1.23610608      less
## 2                AxcR2 / Climb.0 Homog. 0.936177916 -0.02698244      less
## 3                AxcR1 / Climb.1 Homog. 0.065897849  0.20462341      less
## 4                AxcR2 / Climb.1 Homog. 0.061538793  0.04819169      less
## 5                AxcR1 / Erect.0 Homog. 0.065897849  0.20462341      less
## 6                AxcR2 / Erect.0 Homog. 0.061538793  0.04819169      less
## 7                AxcR1 / Erect.1 Homog. 0.839889789 -1.23610608      less
## 8                AxcR2 / Erect.1 Homog. 0.936177916 -0.02698244      less
## 9                AxcR1 / StemS.0 Homog. 0.084930906  1.69877200      less
## 10               AxcR2 / StemS.0 Homog. 0.072328995 -0.35370401      less
## 11               AxcR1 / StemS.1 Homog. 0.677673054  0.66732923      less
## 12               AxcR2 / StemS.1 Homog. 0.683558940  0.69941660      less
## 13               AxcR1 / StemS.2 Homog. 0.141382536 -0.97471883      less
## 14               AxcR2 / StemS.2 Homog. 0.214125410 -0.57182473      less
## 15               AxcR1 / StemA.0 Homog. 0.638291791  1.69089880      less
## 16               AxcR2 / StemA.0 Homog. 0.618650149  0.51642264      less
## 17               AxcR1 / StemA.1 Homog. 0.343059361  1.29931651      less
## 18               AxcR2 / StemA.1 Homog. 0.359912032  1.32198910      less
## 19               AxcR1 / Leave.0 Homog. 0.491428485 -1.75507695      less
## 20               AxcR2 / Leave.0 Homog. 0.556811983  0.56146161      less
## 21               AxcR1 / Leave.1 Homog. 0.508324379  2.30185705      less
## 22               AxcR2 / Leave.1 Homog. 0.428685041  1.22245129      less
## 23       AxcR1 / MaxStemHeight_m      r 0.001987912  0.01365747 two-sided
## 24       AxcR2 / MaxStemHeight_m      r 0.264466379  2.50320737 two-sided
## 25         AxcR1 / MaxStemDia_cm      r 0.120705153  0.93414323 two-sided
## 26         AxcR2 / MaxStemDia_cm      r 0.221464638  2.03959060 two-sided
## 27          AxcR1 / Under.canopy Homog. 0.924832221 -0.19345118      less
## 28          AxcR2 / Under.canopy Homog. 0.944938702  0.75394706      less
## 29     AxcR1 / Under.understorey Homog. 0.075157164  1.09449566      less
## 30     AxcR2 / Under.understorey Homog. 0.051982001 -0.03848357      less
## 31 AxcR1 / AverageFruitLength_cm      r 0.051609120  0.43847120 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r 0.128240151  1.18518730 two-sided
## 33           AxcR1 / Fruit.large Homog. 0.479180668  2.30536632      less
## 34           AxcR2 / Fruit.large Homog. 0.547069452  2.71175647      less
## 35           AxcR1 / Fruit.small Homog. 0.510651838  2.37588150      less
## 36           AxcR2 / Fruit.small Homog. 0.430405233 -0.77704215      less
## 37     AxcR1 / Consp.conspicuous Homog. 0.569272801  1.82539825      less
## 38     AxcR2 / Consp.conspicuous Homog. 0.515477591  0.11928471      less
## 39         AxcR1 / Consp.cryptic Homog. 0.430411253 -0.34889375      less
## 40         AxcR2 / Consp.cryptic Homog. 0.461341932 -0.18664428      less
## 41               AxcR1 / Endem.N Homog. 0.775406254  4.96148018      less
## 42               AxcR2 / Endem.N Homog. 0.623975889 -1.42229751      less
## 43               AxcR1 / Endem.Y Homog. 0.206565714  0.26427154      less
## 44               AxcR2 / Endem.Y Homog. 0.339802139  1.23186183      less
##               Pvalue        Pvalue.adj  
## 1               0.12             0.536  
## 2               0.46 0.778461538461539  
## 3  0.816503800217155            0.9691  
## 4  0.740499457111835            0.9691  
## 5  0.816503800217155            0.9691  
## 6  0.740499457111835            0.9691  
## 7               0.12             0.536  
## 8               0.46 0.778461538461539  
## 9              0.948                 1  
## 10              0.54            0.9691  
## 11             0.716            0.9691  
## 12             0.726            0.9691  
## 13             0.213 0.754285714285714  
## 14             0.336 0.829714285714286  
## 15             0.955                 1  
## 16             0.703 0.893828571428571  
## 17             0.881            0.9691  
## 18             0.863            0.9691  
## 19             0.052            0.4576  
## 20             0.709 0.893828571428571  
## 21             0.993                 1  
## 22             0.857            0.9691  
## 23             0.987                 1  
## 24             0.004             0.132  
## 25              0.38 0.829714285714286  
## 26             0.026              0.33  
## 27             0.355 0.829714285714286  
## 28              0.75 0.916666666666667  
## 29             0.855                 1  
## 30 0.736170212765957            0.9691  
## 31             0.692            0.9691  
## 32             0.266 0.780266666666667  
## 33             0.971 0.993581395348837  
## 34             0.995             0.995  
## 35             0.992                 1  
## 36             0.215             0.473  
## 37             0.972                 1  
## 38              0.53            0.9691  
## 39             0.372 0.829714285714286  
## 40             0.447             0.894  
## 41                 1                 1  
## 42             0.108             0.536  
## 43             0.622            0.9691  
## 44             0.841            0.9691  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

RLQ axes and environmental variables

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                          Test   Stat          Obs    Std.Obs     Alter Pvalue        Pvalue.adj  
## 1        Canopy.Cover / AxcQ1      r -0.005225042 -0.1046551 two-sided  0.904             0.939  
## 2  Understory.Density / AxcQ1      r -0.039109377 -0.4276082 two-sided  0.727             0.912  
## 3         Leaf.Litter / AxcQ1      r -0.011975126 -0.0811434 two-sided  0.939             0.939  
## 4       Soil.Moisture / AxcQ1      r -0.213214982 -1.4974913 two-sided  0.146 0.581333333333333  
## 5                 Cec / AxcQ1      r -0.027025540 -0.3588843 two-sided  0.757             0.912  
## 6                 T50 / AxcQ1      r  0.044494996  0.6684522 two-sided   0.55              0.88  
## 7                 T10 / AxcQ1      r -0.300142333 -1.4469030 two-sided  0.161             0.644  
## 8       Canopy.Height / AxcQ1      r  0.062674241  0.4163836 two-sided   0.76             0.912  
## 9           Elevation / AxcQ1      r  0.075537324  0.4805011 two-sided  0.678             0.912  
## 10      Habit.Primary / AxcQ1 Homog.  0.353990036 -1.3761412      less  0.104             0.624  
## 11    Habit.Secondary / AxcQ1 Homog.  0.574188609  1.0303044      less  0.844 0.920727272727273  
## 12       DIST_TO_EDGE / AxcQ1      r  0.219219415  1.3446398 two-sided  0.188 0.644571428571429  
## 13       Canopy.Cover / AxcQ2      r -0.093816370 -1.2153270 two-sided  0.259            0.6672  
## 14 Understory.Density / AxcQ2      r -0.203506782 -1.7770454 two-sided  0.065              0.39  
## 15        Leaf.Litter / AxcQ2      r  0.151927905  1.1445878 two-sided  0.278            0.6672  
## 16      Soil.Moisture / AxcQ2      r -0.094348556 -0.8294920 two-sided  0.457              0.88  
## 17                Cec / AxcQ2      r  0.054931337  0.8941510 two-sided  0.388 0.846545454545454  
## 18                T50 / AxcQ2      r  0.157686677  1.4317343 two-sided    0.2 0.581333333333333  
## 19                T10 / AxcQ2      r  0.080736423  0.3717995 two-sided  0.695             0.912  
## 20      Canopy.Height / AxcQ2      r -0.104820162 -0.7904038 two-sided  0.492              0.88  
## 21          Elevation / AxcQ2      r  0.239858669  1.5881946 two-sided  0.134            0.6432  
## 22      Habit.Primary / AxcQ2 Homog.  0.555681426  1.0340799      less  0.849 0.926181818181818  
## 23    Habit.Secondary / AxcQ2 Homog.  0.444023222 -0.5910135      less  0.243            0.6672  
## 24       DIST_TO_EDGE / AxcQ2      r -0.110392420 -0.6880360 two-sided  0.528              0.88  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                          Test   Stat          Obs    Std.Obs     Alter Pvalue        Pvalue.adj  
## 1        Canopy.Cover / AxcQ1      r -0.048167244 -1.0619662 two-sided  0.276 0.639272727272727  
## 2  Understory.Density / AxcQ1      r -0.190393597 -1.9784619 two-sided  0.055             0.616  
## 3         Leaf.Litter / AxcQ1      r  0.087803763  0.9900674 two-sided  0.405 0.694285714285714  
## 4       Soil.Moisture / AxcQ1      r  0.162110244  1.5278285 two-sided  0.154             0.616  
## 5                 Cec / AxcQ1      r  0.039013207  0.8259116 two-sided  0.444            0.7104  
## 6                 T50 / AxcQ1      r -0.062618997 -1.5531882 two-sided   0.14             0.616  
## 7                 T10 / AxcQ1      r -0.035164882 -0.3926171 two-sided  0.683 0.885714285714286  
## 8       Canopy.Height / AxcQ1      r  0.166388097  1.3863273 two-sided  0.142             0.616  
## 9           Elevation / AxcQ1      r -0.267643184 -1.7624298 two-sided  0.098             0.616  
## 10      Habit.Primary / AxcQ1 Homog.  0.627039627  5.2677663      less      1                 1  
## 11    Habit.Secondary / AxcQ1 Homog.  0.372950388  0.1609811      less  0.511 0.721411764705882  
## 12       DIST_TO_EDGE / AxcQ1      r -0.142545392 -2.2004544 two-sided   0.02              0.48  
## 13       Canopy.Cover / AxcQ2      r -0.039816375 -1.0777505 two-sided  0.293 0.639272727272727  
## 14 Understory.Density / AxcQ2      r  0.003634773  0.1048194 two-sided  0.912 0.994909090909091  
## 15        Leaf.Litter / AxcQ2      r  0.090431827  1.1069128 two-sided  0.322             0.644  
## 16      Soil.Moisture / AxcQ2      r -0.104713419 -0.9198389 two-sided  0.381 0.694285714285714  
## 17                Cec / AxcQ2      r -0.016003663 -0.2679409 two-sided  0.813 0.886909090909091  
## 18                T50 / AxcQ2      r  0.010094780  0.2037692 two-sided  0.852 0.973714285714286  
## 19                T10 / AxcQ2      r  0.159632657  1.3417132 two-sided  0.216 0.639272727272727  
## 20      Canopy.Height / AxcQ2      r  0.167334976  1.3941289 two-sided  0.185 0.634285714285714  
## 21          Elevation / AxcQ2      r  0.060347040  0.3755701 two-sided  0.745 0.885714285714286  
## 22      Habit.Primary / AxcQ2 Homog.  0.684869599  7.2703433      less      1                 1  
## 23    Habit.Secondary / AxcQ2 Homog.  0.315108243 -0.6261314      less  0.277 0.639272727272727  
## 24       DIST_TO_EDGE / AxcQ2      r -0.020856866 -0.3767687 two-sided  0.738 0.885714285714286  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Results can be represented using a table with colors indicating significance :

## [1] "juveniles"

## [1] "adults"

Significance with axes can also be reported on the factorial map of RLQ analysis. Here, significant associations with the first axis are represented in blue, with the second axis in orange, with both axes in green (variables with no significant association are in black)

## [1] "juveniles"

## [1] "adults"

RUN 4: RLQ/4th corner: Including Luke’s data, broken down by age, converted to Denisities

Run RLQ/ FQ analysis for all lifestage types COMBINED DATA

That’s unreadable, plotting as separate.

## [1] "RLQ for juveniles"

## [1] "RLQ for adults"

Summary of RLQ analysis. How to interpret this?

## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Rjuv, dudiL = Ljuv, dudiQ = Qjuv, scannf = FALSE)
## 
## Total inertia: 0.5071
## 
## Eigenvalues:
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 0.30533 0.11443 0.06096 0.02369 0.00200 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 60.2072 22.5644 12.0197  4.6722  0.3943 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   60.21   82.77   94.79   99.46   99.86 
## 
## (Only 5 dimensions (out of 8) are shown)
## 
## 
## Eigenvalues decomposition:
##         eig     covar      sdR      sdQ      corr
## 1 0.3053304 0.5525671 1.063631 1.508370 0.3444183
## 2 0.1144315 0.3382772 1.111907 1.627049 0.1869837
## 
## Inertia & coinertia R (Rjuv):
##     inertia      max     ratio
## 1  1.131310 1.408984 0.8029265
## 12 2.367647 2.673132 0.8857201
## 
## Inertia & coinertia Q (Qjuv):
##     inertia      max     ratio
## 1  2.275181 3.708939 0.6134317
## 12 4.922468 6.934943 0.7098066
## 
## Correlation L (Ljuv):
##        corr       max     ratio
## 1 0.3444183 0.8781144 0.3922248
## 2 0.1869837 0.8370330 0.2233887
## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Radu, dudiL = Ladu, dudiQ = Qadu, scannf = FALSE)
## 
## Total inertia: 0.3568
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.192475 0.086756 0.067868 0.005586 0.002817 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
## 53.9414 24.3135 19.0201  1.5656  0.7894 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   53.94   78.25   97.27   98.84   99.63 
## 
## (Only 5 dimensions (out of 8) are shown)
## 
## 
## Eigenvalues decomposition:
##          eig     covar      sdR      sdQ      corr
## 1 0.19247487 0.4387196 1.086161 1.913300 0.2111105
## 2 0.08675598 0.2945437 1.102252 1.403553 0.1903882
## 
## Inertia & coinertia R (Radu):
##     inertia      max     ratio
## 1  1.179745 1.387695 0.8501476
## 12 2.394704 2.740164 0.8739273
## 
## Inertia & coinertia Q (Qadu):
##     inertia      max     ratio
## 1  3.660716 4.077184 0.8978541
## 12 5.630678 7.312295 0.7700288
## 
## Correlation L (Ladu):
##        corr       max     ratio
## 1 0.2111105 1.0000000 0.2111105
## 2 0.1903882 0.9480088 0.2008296

Fourth-corner analysis

## [1] "FQ for juveniles"

## [1] "FQ for adults"

With adjusted p-values

## [1] "FQ for Juveniles"
## [1] "FQ for Adults"

Combine both approaches

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs     Std.Obs   Alter Pvalue
## 1 Model 2 0.507133 11.23012548 greater  0.001
## 2 Model 4 0.507133  0.06700219 greater  0.450

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test       Obs    Std.Obs   Alter Pvalue
## 1 Model 2 0.3568223 12.0699667 greater  0.001
## 2 Model 4 0.3568223  0.9624644 greater  0.174

The total inertia of RLQ analysis is equal to the SRLQ multivariate statistic defined in Dray and Legendre (2008). This statistic is returned by the fourthcorner2 function:

## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envCombined, tabL = JuvCombined, tabQ = p_traits2, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 0.507133 
## 
## Based on 943 replicates
## Simulated p-value: 0.4597458 
## Alternative hypothesis: greater 
## 
##     Std.Obs Expectation    Variance 
##  0.04949982  0.49969574  0.02257454
## Monte-Carlo test
## Call: fourthcorner2(tabR = p_envCombined, tabL = AduCombined, tabQ = p_traits2, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 0.3568223 
## 
## Based on 999 replicates
## Simulated p-value: 0.166 
## Alternative hypothesis: greater 
## 
##       Std.Obs   Expectation      Variance 
##  5.000727e-01 -2.399700e+03  2.303439e+07
## [1] "juvenile"

## [1] "adult"

“Another approach is provided by the fourthcorner.rlq function and consists in testing directly the links between RLQ axes and traits (typetest=”Q.axes“) or environmental variables (typetest=”R.axes“).”

RLQ axes and traits

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat          Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.950691650  0.02297862      less
## 2                AxcR2 / Climb.0 Homog.  0.941841026 -0.16365303      less
## 3                AxcR1 / Climb.1 Homog.  0.033731565 -0.11523651      less
## 4                AxcR2 / Climb.1 Homog.  0.049479519  1.45852361      less
## 5                AxcR1 / Erect.0 Homog.  0.033731565 -0.11523651      less
## 6                AxcR2 / Erect.0 Homog.  0.049479519  1.45852361      less
## 7                AxcR1 / Erect.1 Homog.  0.950691650  0.02297862      less
## 8                AxcR2 / Erect.1 Homog.  0.941841026 -0.16365303      less
## 9                AxcR1 / StemS.0 Homog.  0.042486336 -0.62407515      less
## 10               AxcR2 / StemS.0 Homog.  0.059233726  1.32029688      less
## 11               AxcR1 / StemS.1 Homog.  0.600092020  0.06909407      less
## 12               AxcR2 / StemS.1 Homog.  0.663529348  0.98963052      less
## 13               AxcR1 / StemS.2 Homog.  0.337273734  1.95226449      less
## 14               AxcR2 / StemS.2 Homog.  0.267940716 -0.22206955      less
## 15               AxcR1 / StemA.0 Homog.  0.413618760  0.96874178      less
## 16               AxcR2 / StemA.0 Homog.  0.409856391  0.54901906      less
## 17               AxcR1 / StemA.1 Homog.  0.572991793  3.04965858      less
## 18               AxcR2 / StemA.1 Homog.  0.587847629  3.42208062      less
## 19               AxcR1 / Leave.0 Homog.  0.354248320 -0.12272264      less
## 20               AxcR2 / Leave.0 Homog.  0.353383576 -0.08334301      less
## 21               AxcR1 / Leave.1 Homog.  0.616579602  2.85114226      less
## 22               AxcR2 / Leave.1 Homog.  0.646520452  3.38539337      less
## 23       AxcR1 / MaxStemHeight_m      r -0.081641368 -0.51410952 two-sided
## 24       AxcR2 / MaxStemHeight_m      r  0.019080907  0.27036126 two-sided
## 25         AxcR1 / MaxStemDia_cm      r -0.111440071 -0.69917296 two-sided
## 26         AxcR2 / MaxStemDia_cm      r  0.051767377  0.65718792 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.825995793 -0.21400969      less
## 28          AxcR2 / Under.canopy Homog.  0.824299326 -0.29534801      less
## 29     AxcR1 / Under.understorey Homog.  0.131796389 -0.01477115      less
## 30     AxcR2 / Under.understorey Homog.  0.173736036  0.37677877      less
## 31 AxcR1 / AverageFruitLength_cm      r  0.003639056  0.03905259 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r  0.139110582  1.60663803 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.171370100 -0.07021277      less
## 34           AxcR2 / Fruit.large Homog.  0.225979472  0.40238496      less
## 35           AxcR1 / Fruit.small Homog.  0.816271465  3.55292068      less
## 36           AxcR2 / Fruit.small Homog.  0.742787554 -0.65533486      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.665815314  1.43610990      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.699872302  1.90444340      less
## 39         AxcR1 / Consp.cryptic Homog.  0.317460710  0.87173520      less
## 40         AxcR2 / Consp.cryptic Homog.  0.298427569 -0.28843876      less
## 41               AxcR1 / Endem.N Homog.  0.722721789 -0.14631779      less
## 42               AxcR2 / Endem.N Homog.  0.720851064 -0.26717447      less
## 43               AxcR1 / Endem.Y Homog.  0.191637919 -0.40711010      less
## 44               AxcR2 / Endem.Y Homog.  0.272932571  0.59682909      less
##               Pvalue        Pvalue.adj  
## 1              0.342 0.875111111111111  
## 2               0.45 0.875111111111111  
## 3  0.680896478121665 0.881160148157449  
## 4              0.931 0.985395348837209  
## 5  0.680896478121665 0.881160148157449  
## 6              0.931 0.985395348837209  
## 7              0.342 0.875111111111111  
## 8               0.45 0.875111111111111  
## 9              0.345 0.875111111111111  
## 10             0.908 0.985395348837209  
## 11             0.493 0.875111111111111  
## 12             0.855 0.985395348837209  
## 13             0.963 0.985395348837209  
## 14             0.428 0.875111111111111  
## 15             0.824 0.979891891891892  
## 16             0.725 0.935314285714286  
## 17             0.999                 1  
## 18                 1                 1  
## 19             0.461 0.743285714285714  
## 20             0.473 0.743285714285714  
## 21             0.997                 1  
## 22                 1                 1  
## 23             0.649 0.881160148157449  
## 24              0.81 0.963243243243243  
## 25             0.514 0.875111111111111  
## 26             0.535 0.875111111111111  
## 27             0.352 0.875111111111111  
## 28             0.368 0.875111111111111  
## 29             0.568 0.881160148157449  
## 30             0.647 0.881160148157449  
## 31             0.978                 1  
## 32             0.102 0.641142857142857  
## 33             0.537 0.875111111111111  
## 34             0.652 0.881160148157449  
## 35                 1                 1  
## 36             0.245 0.875111111111111  
## 37             0.924                 1  
## 38             0.976                 1  
## 39             0.796 0.972888888888889  
## 40             0.396           0.69696  
## 41             0.422 0.875111111111111  
## 42             0.394 0.875111111111111  
## 43             0.389 0.875111111111111  
## 44             0.744 0.935314285714286  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat         Obs      Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.97179858  0.914152396      less
## 2                AxcR2 / Climb.0 Homog.  0.97948921  2.604089304      less
## 3                AxcR1 / Climb.1 Homog.  0.02531189 -0.352455173      less
## 4                AxcR2 / Climb.1 Homog.  0.01348188 -0.589989143      less
## 5                AxcR1 / Erect.0 Homog.  0.02531189 -0.352455173      less
## 6                AxcR2 / Erect.0 Homog.  0.01348188 -0.589989143      less
## 7                AxcR1 / Erect.1 Homog.  0.97179858  0.914152396      less
## 8                AxcR2 / Erect.1 Homog.  0.97948921  2.604089304      less
## 9                AxcR1 / StemS.0 Homog.  0.04679292 -0.460331662      less
## 10               AxcR2 / StemS.0 Homog.  0.04346445 -0.591578392      less
## 11               AxcR1 / StemS.1 Homog.  0.47268434 -0.679161415      less
## 12               AxcR2 / StemS.1 Homog.  0.53370050  3.049384513      less
## 13               AxcR1 / StemS.2 Homog.  0.46353583  0.994016377      less
## 14               AxcR2 / StemS.2 Homog.  0.41647109  0.851510796      less
## 15               AxcR1 / StemA.0 Homog.  0.21466306 -1.191998737      less
## 16               AxcR2 / StemA.0 Homog.  0.29338290  3.790097450      less
## 17               AxcR1 / StemA.1 Homog.  0.78478607  1.201653576      less
## 18               AxcR2 / StemA.1 Homog.  0.69891091  3.317531288      less
## 19               AxcR1 / Leave.0 Homog.  0.18698199 -0.871186511      less
## 20               AxcR2 / Leave.0 Homog.  0.26528813  4.371985909      less
## 21               AxcR1 / Leave.1 Homog.  0.81301791  0.915403428      less
## 22               AxcR2 / Leave.1 Homog.  0.71751015  2.856201824      less
## 23       AxcR1 / MaxStemHeight_m      r -0.11128831 -1.473647834 two-sided
## 24       AxcR2 / MaxStemHeight_m      r -0.04326206 -0.598501338 two-sided
## 25         AxcR1 / MaxStemDia_cm      r -0.16305380 -2.163140637 two-sided
## 26         AxcR2 / MaxStemDia_cm      r -0.02330345 -0.293644047 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.84135842 -0.061285857      less
## 28          AxcR2 / Under.canopy Homog.  0.84882465  0.912217124      less
## 29     AxcR1 / Under.understorey Homog.  0.15371029  0.063375226      less
## 30     AxcR2 / Under.understorey Homog.  0.14529232 -0.001334491      less
## 31 AxcR1 / AverageFruitLength_cm      r -0.20867857 -2.765443755 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r  0.03378664  0.527096705 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.21701516  0.134072032      less
## 34           AxcR2 / Fruit.large Homog.  0.22896442  0.253936793      less
## 35           AxcR1 / Fruit.small Homog.  0.74297350 -0.364222176      less
## 36           AxcR2 / Fruit.small Homog.  0.76496435 -0.263594037      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.76188172  1.391075196      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.72621006  1.411520681      less
## 39         AxcR1 / Consp.cryptic Homog.  0.23058913 -0.512603616      less
## 40         AxcR2 / Consp.cryptic Homog.  0.27337586  1.916362313      less
## 41               AxcR1 / Endem.N Homog.  0.83942871  0.420995017      less
## 42               AxcR2 / Endem.N Homog.  0.85125177  1.396776241      less
## 43               AxcR1 / Endem.Y Homog.  0.15572888 -0.514687259      less
## 44               AxcR2 / Endem.Y Homog.  0.13859593 -0.751428706      less
##               Pvalue        Pvalue.adj  
## 1              0.819                 1  
## 2              0.999                 1  
## 3   0.50308261405672 0.870941176470588  
## 4  0.316892725030826 0.870941176470588  
## 5   0.50308261405672 0.870941176470588  
## 6  0.316892725030826 0.870941176470588  
## 7              0.819                 1  
## 8              0.999                 1  
## 9  0.425334706488157 0.870941176470588  
## 10 0.377960865087539 0.870941176470588  
## 11             0.305 0.870941176470588  
## 12                 1                 1  
## 13              0.78  0.91821052631579  
## 14             0.793  0.91821052631579  
## 15             0.112 0.289882352941176  
## 16                 1                 1  
## 17             0.893                 1  
## 18                 1                 1  
## 19             0.191 0.442315789473684  
## 20                 1                 1  
## 21             0.823                 1  
## 22                 1                 1  
## 23             0.155 0.757777777777778  
## 24             0.569 0.870941176470588  
## 25              0.02 0.146666666666667  
## 26              0.77  0.91821052631579  
## 27             0.324 0.870941176470588  
## 28              0.82                 1  
## 29 0.670670670670671 0.870941176470588  
## 30 0.607607607607608 0.870941176470588  
## 31             0.002            0.0176 *
## 32             0.581 0.870941176470588  
## 33             0.673 0.870941176470588  
## 34             0.655 0.870941176470588  
## 35             0.254 0.870941176470588  
## 36             0.343 0.870941176470588  
## 37             0.924                 1  
## 38             0.924                 1  
## 39             0.321 0.614086956521739  
## 40             0.962                 1  
## 41             0.648             0.891  
## 42              0.92                 1  
## 43             0.388 0.870941176470588  
## 44             0.266 0.870941176470588  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

RLQ axes and environmental variables

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                     Test   Stat          Obs      Std.Obs     Alter Pvalue        Pvalue.adj  
## 1    Under.dense / AxcQ1 Homog.  0.221464258  0.327772208      less  0.743            0.8916  
## 2      Under.low / AxcQ1 Homog.  0.218403774 -0.813982319      less  0.191           0.42975  
## 3   Under.medium / AxcQ1 Homog.  0.550063386  1.365268845      less  0.923             0.997  
## 4            Cec / AxcQ1      r  0.042828674  1.293053664 two-sided  0.123 0.339428571428571  
## 5            T50 / AxcQ1      r -0.000714878  0.007297674 two-sided  0.997             0.997  
## 6            T10 / AxcQ1      r -0.026242242 -0.815171179 two-sided  0.397            0.7146  
## 7  Canopy.Height / AxcQ1      r -0.043252066 -0.639468467 two-sided   0.54 0.694285714285714  
## 8      Elevation / AxcQ1      r  0.343316027  2.188200528 two-sided  0.009             0.108  
## 9   DIST_TO_EDGE / AxcQ1      r  0.043264696  0.712888388 two-sided  0.506 0.694285714285714  
## 10   Under.dense / AxcQ2 Homog.  0.271414040  2.285892891      less  0.978             0.997  
## 11     Under.low / AxcQ2 Homog.  0.208125884 -1.563276630      less  0.061             0.216  
## 12  Under.medium / AxcQ2 Homog.  0.513648363  0.136618803      less  0.501            0.7515  
## 13           Cec / AxcQ2      r -0.043906956 -1.249087138 two-sided  0.132 0.339428571428571  
## 14           T50 / AxcQ2      r -0.017580374 -0.500141307 two-sided  0.554 0.767076923076923  
## 15           T10 / AxcQ2      r  0.024502707  0.711369310 two-sided  0.464            0.7515  
## 16 Canopy.Height / AxcQ2      r  0.153488306  2.266352612 two-sided  0.012             0.108  
## 17     Elevation / AxcQ2      r  0.019284269  0.162401495 two-sided  0.894             0.986  
## 18  DIST_TO_EDGE / AxcQ2      r  0.098216141  1.802778491 two-sided  0.071             0.216  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                     Test   Stat          Obs     Std.Obs     Alter Pvalue        Pvalue.adj  
## 1    Under.dense / AxcQ1 Homog.  0.214360348 -0.03128674      less  0.502 0.695076923076923  
## 2      Under.low / AxcQ1 Homog.  0.306199419  1.62081624      less  0.974             0.974  
## 3   Under.medium / AxcQ1 Homog.  0.476937675 -1.30113594      less  0.103            0.3708  
## 4            Cec / AxcQ1      r  0.031433924  1.12053990 two-sided  0.264             0.396  
## 5            T50 / AxcQ1      r -0.007846279 -0.27933305 two-sided  0.779          0.876375  
## 6            T10 / AxcQ1      r -0.132361714 -1.99303642 two-sided   0.01             0.066 .
## 7  Canopy.Height / AxcQ1      r -0.152381058 -1.91431680 two-sided  0.011             0.066 .
## 8      Elevation / AxcQ1      r  0.084566234  1.25240235 two-sided  0.224             0.504  
## 9   DIST_TO_EDGE / AxcQ1      r -0.033692825 -0.45611548 two-sided   0.73             0.846  
## 10   Under.dense / AxcQ2 Homog.  0.233712304  0.57698290      less  0.745          0.876375  
## 11     Under.low / AxcQ2 Homog.  0.286539392  0.38202529      less   0.66 0.848571428571429  
## 12  Under.medium / AxcQ2 Homog.  0.476587859 -1.02035920      less  0.167 0.429428571428571  
## 13           Cec / AxcQ2      r -0.005443898 -0.16437119 two-sided  0.854             0.899  
## 14           T50 / AxcQ2      r -0.030475271 -0.94304199 two-sided  0.352            0.6336  
## 15           T10 / AxcQ2      r  0.036422985  0.53148379 two-sided  0.675             0.846  
## 16 Canopy.Height / AxcQ2      r  0.050444774  0.62290993 two-sided  0.613             0.846  
## 17     Elevation / AxcQ2      r  0.175774038  2.57900815 two-sided  0.006             0.066 .
## 18  DIST_TO_EDGE / AxcQ2      r  0.071782890  1.05208457 two-sided  0.336            0.6336  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Results can be represented using a table with colors indicating significance :

## [1] "juveniles"

## [1] "adults"

Significance with axes can also be reported on the factorial map of RLQ analysis. Here, significant associations with the first axis are represented in blue, with the second axis in orange, with both axes in green (variables with no significant association are in black)

## [1] "juveniles"

## [1] "adults"

RUN 5: Only Luke’s data as abundances

## 'data.frame':    19 obs. of  12 variables:
##  $ Climbing             : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 2 1 ...
##  $ Erect                : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 1 2 ...
##  $ StemSolitary         : Factor w/ 3 levels "0","1","2": 2 2 2 3 3 3 2 2 1 3 ...
##  $ StemArmed            : Factor w/ 2 levels "0","1": 2 2 1 2 2 2 1 1 1 1 ...
##  $ LeavesArmed          : Factor w/ 2 levels "0","1": 2 2 1 2 2 2 1 1 2 1 ...
##  $ MaxStemHeight_m      : num  4.5 15 35 10 18 10 10 4.5 20 3 ...
##  $ MaxStemDia_cm        : num  6 30 60 8 25 10 8 3 3 3.4 ...
##  $ UnderstoreyCanopy    : Factor w/ 2 levels "canopy","understorey": 2 1 1 1 1 1 1 2 1 2 ...
##  $ AverageFruitLength_cm: num  0.75 5.5 5 2 5 ...
##  $ FruitSizeCategorical : Factor w/ 2 levels "large","small": 2 1 1 2 1 2 2 2 2 2 ...
##  $ Conspicuousness      : Factor w/ 2 levels "conspicuous",..: 2 1 2 1 1 1 1 2 1 2 ...
##  $ Endemic              : Factor w/ 2 levels "N","Y": 2 1 2 1 1 1 1 1 1 1 ...
## 'data.frame':    2300 obs. of  7 variables:
##  $ ELEV        : num  414 413 412 411 411 409 407 406 405 405 ...
##  $ DIST_TO_EDGE: num  1.5 6 11.4 16.6 23.3 28.5 33.5 38.2 40.6 44.2 ...
##  $ CANOPY      : num  25 25 31 28 25 23 7 21 25 20 ...
##  $ TEN         : num  5 3 4 3 5 5 4 4 4 5 ...
##  $ FIFTY       : num  0 0 0 1 0 0 0 0 0 0 ...
##  $ CECR        : num  0 0 0 0 0 0 0 0 1 0 ...
##  $ UNDERSTORY  : Factor w/ 3 levels "dense","low",..: 1 1 1 1 1 1 1 3 3 3 ...
## 'data.frame':    2300 obs. of  19 variables:
##  $ AST  : num  0 0 0 0 1 0 0 0 0 0 ...
##  $ AT   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ ATT  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ BATCO: num  0 0 0 0 0 0 0 3 0 0 ...
##  $ BATSE: num  0 0 0 0 0 0 0 0 0 0 ...
##  $ BG   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ CHAM : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ CP   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ DESM : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ IR   : num  0 0 1 0 1 0 0 0 2 0 ...
##  $ ONE  : num  0 2 0 2 2 0 0 0 0 0 ...
##  $ PHD  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ PHOL : num  1 0 2 0 0 0 0 0 0 0 ...
##  $ PRDE : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ SOC  : num  0 0 0 0 0 0 0 1 0 1 ...
##  $ SYN  : num  0 0 1 0 0 0 0 1 0 0 ...
##  $ TAG  : num  0 0 1 3 2 0 0 0 0 0 ...
##  $ WET  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ GO   : num  0 0 1 0 1 0 0 0 1 0 ...
## 'data.frame':    2300 obs. of  18 variables:
##  $ AST  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ AT   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ BATCO: num  0 0 0 0 0 0 0 0 0 0 ...
##  $ BATSE: num  0 0 0 0 0 0 0 0 0 0 ...
##  $ BG   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ CHAM : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ CP   : num  0 0 0 1 0 0 0 0 0 0 ...
##  $ DESM : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ IR   : num  0 0 0 0 0 0 0 0 0 1 ...
##  $ ONE  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ PHD  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ PHOL : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ PRDE : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ SOC  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ SYN  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ TAG  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ WET  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ GO   : num  0 0 0 0 0 0 0 0 0 0 ...
##       Climbing Erect StemSolitary StemArmed LeavesArmed MaxStemHeight_m
## AT           0     1            1         1           1             4.5
## AST          0     1            1         1           1            15.0
## BATCO        0     1            2         1           1            10.0
## BG           0     1            2         1           1            18.0
## BATSE        0     1            2         1           1            10.0
## CHAM         0     1            1         0           0            10.0
## CP           0     1            1         0           0             4.5
## DESM         1     0            0         0           1            20.0
## GO           0     1            2         0           0             3.0
## IR           0     1            1         0           0            30.0
## ONE          0     1            1         0           0            26.0
## PHD          0     1            1         0           0            10.0
## PHOL         0     1            1         0           0            12.0
## TAG          0     1            1         0           0            15.0
## PRDE         0     1            2         0           0            10.0
## SOC          0     1            1         0           0            20.0
## SYN          0     1            0         0           0             6.0
## WET          0     1            1         0           0            10.0
##       MaxStemDia_cm UnderstoreyCanopy AverageFruitLength_cm
## AT              6.0       understorey                 0.750
## AST            30.0            canopy                 5.500
## BATCO           8.0            canopy                 2.000
## BG             25.0            canopy                 5.000
## BATSE          10.0            canopy                 1.900
## CHAM            8.0            canopy                 1.650
## CP              3.0       understorey                 1.250
## DESM            3.0            canopy                 1.823
## GO              3.4       understorey                 0.760
## IR             70.0            canopy                 2.350
## ONE            45.0            canopy                 3.500
## PHD            12.0            canopy                 1.215
## PHOL           22.0            canopy                 1.385
## TAG            30.0            canopy                 7.500
## PRDE           12.0            canopy                 0.900
## SOC            20.0            canopy                 3.000
## SYN             5.0            canopy                 2.350
## WET            13.0            canopy                 2.500
##       FruitSizeCategorical Conspicuousness Endemic
## AT                   small         cryptic       Y
## AST                  large     conspicuous       N
## BATCO                small     conspicuous       N
## BG                   large     conspicuous       N
## BATSE                small     conspicuous       N
## CHAM                 small     conspicuous       N
## CP                   small         cryptic       N
## DESM                 small     conspicuous       N
## GO                   small         cryptic       N
## IR                   small     conspicuous       N
## ONE                  small         cryptic       N
## PHD                  small     conspicuous       Y
## PHOL                 small         cryptic       N
## TAG                  large         cryptic       Y
## PRDE                 small         cryptic       N
## SOC                  small         cryptic       N
## SYN                  small     conspicuous       N
## WET                  small     conspicuous       Y

Run RLQ/ FQ analysis for all lifestage types COMBINED DATA

That’s unreadable, plotting as separate.

## [1] "RLQ for juveniles"

## [1] "RLQ for adults"

Summary of RLQ analysis. How to interpret this?

## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Rjuv, dudiL = Ljuv, dudiQ = Qjuv, scannf = FALSE)
## 
## Total inertia: 0.3496
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.253201 0.051745 0.029468 0.009449 0.002748 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
##  72.431  14.802   8.430   2.703   0.786 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   72.43   87.23   95.66   98.37   99.15 
## 
## (Only 5 dimensions (out of 8) are shown)
## 
## 
## Eigenvalues decomposition:
##          eig     covar      sdR      sdQ      corr
## 1 0.25320147 0.5031913 1.102316 1.462761 0.3120711
## 2 0.05174511 0.2274755 1.017318 1.356392 0.1648515
## 
## Inertia & coinertia R (Rjuv):
##     inertia      max     ratio
## 1  1.215101 1.427770 0.8510478
## 12 2.250036 2.706504 0.8313439
## 
## Inertia & coinertia Q (Qjuv):
##     inertia      max     ratio
## 1  2.139670 3.800487 0.5629991
## 12 3.979469 6.041936 0.6586413
## 
## Correlation L (Ljuv):
##        corr       max     ratio
## 1 0.3120711 0.8790003 0.3550296
## 2 0.1648515 0.8158605 0.2020585
## RLQ analysis
## 
## Class: rlq dudi
## Call: rlq(dudiR = Radu, dudiL = Ladu, dudiQ = Qadu, scannf = FALSE)
## 
## Total inertia: 0.2272
## 
## Eigenvalues:
##      Ax1      Ax2      Ax3      Ax4      Ax5 
## 0.112675 0.083210 0.014000 0.008495 0.006291 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
##  49.583  36.617   6.161   3.738   2.768 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   49.58   86.20   92.36   96.10   98.87 
## 
## (Only 5 dimensions (out of 8) are shown)
## 
## 
## Eigenvalues decomposition:
##          eig     covar       sdR      sdQ      corr
## 1 0.11267484 0.3356707 0.9879998 1.643250 0.2067535
## 2 0.08320955 0.2884607 1.1240015 1.863746 0.1376997
## 
## Inertia & coinertia R (Radu):
##      inertia      max     ratio
## 1  0.9761436 1.532974 0.6367647
## 12 2.2395230 2.712636 0.8255891
## 
## Inertia & coinertia Q (Qadu):
##     inertia      max     ratio
## 1  2.700272 4.108066 0.6573099
## 12 6.173821 6.921196 0.8920166
## 
## Correlation L (Ladu):
##        corr       max     ratio
## 1 0.2067535 0.9773324 0.2115488
## 2 0.1376997 0.9603342 0.1433873

Fourth-corner analysis

## [1] "FQ for juveniles"

## [1] "FQ for adults"

With adjusted p-values

## [1] "FQ for Juveniles"
## [1] "FQ for Adults"

Combine both approaches

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test       Obs   Std.Obs   Alter Pvalue
## 1 Model 2 0.3495769  5.294913 greater  0.003
## 2 Model 4 0.3495769 -1.001535 greater  0.841

## class: krandtest lightkrandtest 
## Monte-Carlo tests
## Call: randtest.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6)
## 
## Number of tests:   2 
## 
## Adjustment method for multiple comparisons:   none 
## Permutation number:   999 
##      Test      Obs    Std.Obs   Alter Pvalue
## 1 Model 2 0.227245  2.1173563 greater  0.032
## 2 Model 4 0.227245 -0.8647113 greater  0.809

The total inertia of RLQ analysis is equal to the SRLQ multivariate statistic defined in Dray and Legendre (2008). This statistic is returned by the fourthcorner2 function:

## Monte-Carlo test
## Call: fourthcorner2(tabR = tempLukeENV, tabL = tempLukeP_juv, tabQ = p_traits3, 
##     modeltype = 6, nrepet = nrepet, p.adjust.method.G = "fdr")
## 
## Observation: 0.3495769 
## 
## Based on 999 replicates
## Simulated p-value: 0.837 
## Alternative hypothesis: greater 
## 
##     Std.Obs Expectation    Variance 
##  -0.9950583   0.5070412   0.0250419
## Monte-Carlo test
## Call: fourthcorner2(tabR = tempLukeENV, tabL = tempLukeP_ad, tabQ = subset(p_traits3, 
##     rownames(p_traits3) != "ATT"), modeltype = 6, nrepet = nrepet, 
##     p.adjust.method.G = "fdr")
## 
## Observation: 0.227245 
## 
## Based on 999 replicates
## Simulated p-value: 0.782 
## Alternative hypothesis: greater 
## 
##      Std.Obs  Expectation     Variance 
## -0.830801471  0.277645441  0.003680229
## [1] "juvenile"

## [1] "adult"

“Another approach is provided by the fourthcorner.rlq function and consists in testing directly the links between RLQ axes and traits (typetest=”Q.axes“) or environmental variables (typetest=”R.axes“).”

RLQ axes and traits

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat           Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.9987687707  0.88682739      less
## 2                AxcR2 / Climb.0 Homog.  0.9987476602  0.99016069      less
## 3                AxcR1 / Climb.1 Homog.  0.0008108668 -0.32899201      less
## 4                AxcR2 / Climb.1 Homog.  0.0003319984 -0.98527357      less
## 5                AxcR1 / Erect.0 Homog.  0.0008108668 -0.32899201      less
## 6                AxcR2 / Erect.0 Homog.  0.0003319984 -0.98527357      less
## 7                AxcR1 / Erect.1 Homog.  0.9987687707  0.88682739      less
## 8                AxcR2 / Erect.1 Homog.  0.9987476602  0.99016069      less
## 9                AxcR1 / StemS.0 Homog.  0.0450451117 -0.61118930      less
## 10               AxcR2 / StemS.0 Homog.  0.0476509254 -0.79087050      less
## 11               AxcR1 / StemS.1 Homog.  0.7430481116  2.76091522      less
## 12               AxcR2 / StemS.1 Homog.  0.7086722442  0.69530715      less
## 13               AxcR1 / StemS.2 Homog.  0.1889971996 -0.53878978      less
## 14               AxcR2 / StemS.2 Homog.  0.2372120489 -0.24498835      less
## 15               AxcR1 / StemA.0 Homog.  0.9512348914  2.69794755      less
## 16               AxcR2 / StemA.0 Homog.  0.9558284108  2.64687294      less
## 17               AxcR1 / StemA.1 Homog.  0.0456234804 -1.88476111      less
## 18               AxcR2 / StemA.1 Homog.  0.0427127561 -2.18423441      less
## 19               AxcR1 / Leave.0 Homog.  0.9500246343  2.66787336      less
## 20               AxcR2 / Leave.0 Homog.  0.9545544131  2.61283954      less
## 21               AxcR1 / Leave.1 Homog.  0.0472049444 -2.24577836      less
## 22               AxcR2 / Leave.1 Homog.  0.0436668939 -2.58334353      less
## 23       AxcR1 / MaxStemHeight_m      r -0.0056327444 -0.10883613 two-sided
## 24       AxcR2 / MaxStemHeight_m      r -0.1036183894 -1.62913691 two-sided
## 25         AxcR1 / MaxStemDia_cm      r  0.0585498660  0.43073260 two-sided
## 26         AxcR2 / MaxStemDia_cm      r -0.1606342290 -2.95547643 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.8734630866  3.53595591      less
## 28          AxcR2 / Under.canopy Homog.  0.8480620518  0.86129364      less
## 29     AxcR1 / Under.understorey Homog.  0.1260046718 -0.14715570      less
## 30     AxcR2 / Under.understorey Homog.  0.1507807032 -0.08355338      less
## 31 AxcR1 / AverageFruitLength_cm      r -0.2271876191 -1.34660127 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r -0.0026733621 -0.06479005 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.1822455527 -0.08507815      less
## 34           AxcR2 / Fruit.large Homog.  0.1412713332 -0.71455933      less
## 35           AxcR1 / Fruit.small Homog.  0.7479025293 -0.33449854      less
## 36           AxcR2 / Fruit.small Homog.  0.8584962461  2.26104800      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.3201754800  0.06716201      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.3921603600  2.86361708      less
## 39         AxcR1 / Consp.cryptic Homog.  0.6588325370  1.59024693      less
## 40         AxcR2 / Consp.cryptic Homog.  0.6074045296  1.16697737      less
## 41               AxcR1 / Endem.N Homog.  0.6367268132 -0.85639175      less
## 42               AxcR2 / Endem.N Homog.  0.7368704921  2.45061784      less
## 43               AxcR1 / Endem.Y Homog.  0.3310067021  0.85285614      less
## 44               AxcR2 / Endem.Y Homog.  0.2630461434  0.03652826      less
##    Pvalue         Pvalue.adj  
## 1   0.952                  1  
## 2   0.884                  1  
## 3   0.458  0.719714285714286  
## 4   0.049              0.308  
## 5   0.458  0.719714285714286  
## 6   0.049              0.308  
## 7   0.952                  1  
## 8   0.884                  1  
## 9   0.373  0.863789473684211  
## 10  0.298             0.8195  
## 11  0.992                  1  
## 12  0.738                  1  
## 13  0.325  0.823777777777778  
## 14  0.433  0.925692307692308  
## 15  0.998                  1  
## 16  0.996                  1  
## 17  0.004              0.044 *
## 18  0.005              0.044 *
## 19  0.999                  1  
## 20  0.996                  1  
## 21  0.001              0.022 *
## 22  0.002 0.0293333333333333 *
## 23  0.926                  1  
## 24   0.09               0.44  
## 25  0.704                  1  
## 26  0.007             0.0308 *
## 27      1                  1  
## 28  0.821                  1  
## 29  0.508  0.925692307692308  
## 30  0.491  0.925692307692308  
## 31  0.188  0.636307692307692  
## 32  0.945                  1  
## 33  0.523  0.925692307692308  
## 34  0.249             0.7304  
## 35  0.337  0.823777777777778  
## 36      1                  1  
## 37  0.552  0.809032258064516  
## 38  0.989                  1  
## 39  0.937                  1  
## 40  0.878                  1  
## 41  0.212  0.666285714285714  
## 42  0.999                  1  
## 43  0.814                  1  
## 44  0.547  0.925692307692308  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "Q.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                             Test   Stat           Obs     Std.Obs     Alter
## 1                AxcR1 / Climb.0 Homog.  0.9991347824  0.65487938      less
## 2                AxcR2 / Climb.0 Homog.  0.9986996548  0.61722337      less
## 3                AxcR1 / Climb.1 Homog.  0.0003177235 -0.44855971      less
## 4                AxcR2 / Climb.1 Homog.  0.0005084710 -0.22805059      less
## 5                AxcR1 / Erect.0 Homog.  0.0003177235 -0.44855971      less
## 6                AxcR2 / Erect.0 Homog.  0.0005084710 -0.22805059      less
## 7                AxcR1 / Erect.1 Homog.  0.9991347824  0.65487938      less
## 8                AxcR2 / Erect.1 Homog.  0.9986996548  0.61722337      less
## 9                AxcR1 / StemS.0 Homog.  0.0213369430 -0.71889219      less
## 10               AxcR2 / StemS.0 Homog.  0.0276319001 -0.63837652      less
## 11               AxcR1 / StemS.1 Homog.  0.6665462592  0.33491920      less
## 12               AxcR2 / StemS.1 Homog.  0.6331971702  0.14002055      less
## 13               AxcR1 / StemS.2 Homog.  0.2845234073  0.12382590      less
## 14               AxcR2 / StemS.2 Homog.  0.3202676786  0.32619963      less
## 15               AxcR1 / StemA.0 Homog.  0.9102057866  1.14443704      less
## 16               AxcR2 / StemA.0 Homog.  0.9211187280  1.14962293      less
## 17               AxcR1 / StemA.1 Homog.  0.0611473154 -1.25096409      less
## 18               AxcR2 / StemA.1 Homog.  0.0779316181  0.07817659      less
## 19               AxcR1 / Leave.0 Homog.  0.9094298340  1.41392210      less
## 20               AxcR2 / Leave.0 Homog.  0.9197977076  1.39764104      less
## 21               AxcR1 / Leave.1 Homog.  0.0635809214 -1.04251740      less
## 22               AxcR2 / Leave.1 Homog.  0.0790007011  0.04882859      less
## 23       AxcR1 / MaxStemHeight_m      r -0.0624278693 -0.59096454 two-sided
## 24       AxcR2 / MaxStemHeight_m      r  0.0884008818  0.93645805 two-sided
## 25         AxcR1 / MaxStemDia_cm      r -0.0242877015 -0.15589571 two-sided
## 26         AxcR2 / MaxStemDia_cm      r  0.0927305433  1.03739751 two-sided
## 27          AxcR1 / Under.canopy Homog.  0.8111248473  0.96310258      less
## 28          AxcR2 / Under.canopy Homog.  0.7772226093 -0.40410675      less
## 29     AxcR1 / Under.understorey Homog.  0.1868817567  0.24621328      less
## 30     AxcR2 / Under.understorey Homog.  0.2116166257  0.39975532      less
## 31 AxcR1 / AverageFruitLength_cm      r -0.0469149117 -0.35465609 two-sided
## 32 AxcR2 / AverageFruitLength_cm      r  0.0652052957  0.77729753 two-sided
## 33           AxcR1 / Fruit.large Homog.  0.0914789977 -0.46226195      less
## 34           AxcR2 / Fruit.large Homog.  0.1021589679 -0.36813505      less
## 35           AxcR1 / Fruit.small Homog.  0.8940530723  0.43263672      less
## 36           AxcR2 / Fruit.small Homog.  0.8968200109  0.42873919      less
## 37     AxcR1 / Consp.conspicuous Homog.  0.5144121459 -0.22763547      less
## 38     AxcR2 / Consp.conspicuous Homog.  0.5412733470 -0.08087569      less
## 39         AxcR1 / Consp.cryptic Homog.  0.4832004272  0.27520636      less
## 40         AxcR2 / Consp.cryptic Homog.  0.4511795915  0.09528052      less
## 41               AxcR1 / Endem.N Homog.  0.5800938861  1.29634665      less
## 42               AxcR2 / Endem.N Homog.  0.5403177585 -1.46739872      less
## 43               AxcR1 / Endem.Y Homog.  0.4143389765  1.32691247      less
## 44               AxcR2 / Endem.Y Homog.  0.4568520585  1.52039066      less
##    Pvalue        Pvalue.adj  
## 1       1                 1  
## 2       1                 1  
## 3   0.464 0.729142857142857  
## 4   0.585 0.745684210526316  
## 5   0.464 0.729142857142857  
## 6   0.585 0.745684210526316  
## 7       1                 1  
## 8       1                 1  
## 9   0.234             0.792  
## 10  0.295 0.920857142857143  
## 11   0.51 0.920857142857143  
## 12  0.442 0.920857142857143  
## 13  0.666                 1  
## 14  0.713                 1  
## 15  0.933                 1  
## 16  0.942                 1  
## 17  0.045             0.528  
## 18  0.601 0.745684210526316  
## 19  0.983                 1  
## 20  0.985                 1  
## 21  0.141 0.504307692307692  
## 22  0.591 0.745684210526316  
## 23  0.586 0.920857142857143  
## 24  0.382 0.920857142857143  
## 25  0.876                 1  
## 26  0.331 0.920857142857143  
## 27  0.834 0.873714285714286  
## 28  0.196             0.792  
## 29  0.771                 1  
## 30  0.798                 1  
## 31  0.759                 1  
## 32  0.489 0.920857142857143  
## 33  0.442 0.920857142857143  
## 34  0.496 0.920857142857143  
## 35  0.563 0.920857142857143  
## 36  0.552 0.920857142857143  
## 37  0.423 0.920857142857143  
## 38   0.43 0.920857142857143  
## 39  0.584 0.920857142857143  
## 40   0.57 0.920857142857143  
## 41  0.895             0.895  
## 42  0.153             0.748  
## 43  0.818                 1  
## 44  0.858                 1  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

RLQ axes and environmental variables

## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.juv, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                    Test   Stat           Obs      Std.Obs     Alter Pvalue        Pvalue.adj  
## 1          ELEV / AxcQ1      r  2.754935e-01  1.743928199 two-sided  0.066             0.231  
## 2  DIST_TO_EDGE / AxcQ1      r -1.760315e-05  0.007077008 two-sided  0.993             0.993  
## 3        CANOPY / AxcQ1      r -9.890480e-02 -1.549484414 two-sided  0.116            0.4212  
## 4           TEN / AxcQ1      r -3.881277e-02 -0.655351763 two-sided  0.504             0.756  
## 5         FIFTY / AxcQ1      r -3.845224e-02 -0.594885654 two-sided  0.379            0.7218  
## 6          CECR / AxcQ1      r  2.706638e-02  0.474041392 two-sided  0.282            0.7218  
## 7   UNDER.dense / AxcQ1 Homog.  1.502737e-01 -0.336376590      less   0.59 0.816923076923077  
## 8     UNDER.low / AxcQ1 Homog.  2.036597e-01 -0.525871750      less  0.401            0.7218  
## 9  UNDER.medium / AxcQ1 Homog.  6.171264e-01  1.702212261      less  0.963             0.974  
## 10         ELEV / AxcQ2      r  1.925545e-02  0.133979858 two-sided  0.901             0.974  
## 11 DIST_TO_EDGE / AxcQ2      r -1.362020e-01 -2.313959099 two-sided  0.021             0.231  
## 12       CANOPY / AxcQ2      r -5.509804e-02 -1.037127674 two-sided  0.325            0.7218  
## 13          TEN / AxcQ2      r  1.399334e-02  0.300300278 two-sided  0.742             0.954  
## 14        FIFTY / AxcQ2      r  9.046060e-03  0.149636638 two-sided  0.851          0.957375  
## 15         CECR / AxcQ2      r  4.142150e-02  0.847226853 two-sided  0.117            0.4212  
## 16  UNDER.dense / AxcQ2 Homog.  3.018460e-01  2.340105936      less  0.991             0.993  
## 17    UNDER.low / AxcQ2 Homog.  1.929088e-01 -0.974656749      less  0.064             0.384  
## 18 UNDER.medium / AxcQ2 Homog.  5.010706e-01 -0.933944686      less  0.298            0.7218  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Fourth-corner Statistics
## ------------------------
## Permutation method  Comb. 2 and 4  ( 999  permutations)
## 
## Adjustment method for multiple comparisons:   fdr 
## call:  fourthcorner.rlq(xtest = rlq.adu, nrepet = nrepet, modeltype = 6,      typetest = "R.axes", p.adjust.method.G = "fdr", p.adjust.method.D = "fdr") 
## 
## ---
## 
##                    Test   Stat         Obs     Std.Obs     Alter Pvalue        Pvalue.adj  
## 1          ELEV / AxcQ1      r  0.14022339  1.32154698 two-sided  0.202             0.434  
## 2  DIST_TO_EDGE / AxcQ1      r  0.01288337  0.33631748 two-sided  0.779 0.842823529411765  
## 3        CANOPY / AxcQ1      r  0.05280187  0.85151475 two-sided  0.434 0.646714285714286  
## 4           TEN / AxcQ1      r  0.07951111  1.86534124 two-sided  0.045            0.3744  
## 5         FIFTY / AxcQ1      r  0.04350150  1.15331139 two-sided  0.256            0.4608  
## 6          CECR / AxcQ1      r -0.04232425 -1.14604755 two-sided  0.249            0.4608  
## 7   UNDER.dense / AxcQ1 Homog.  0.17355089 -0.05312864      less  0.489 0.676285714285714  
## 8     UNDER.low / AxcQ1 Homog.  0.28645051  1.92488001      less  0.965             0.965  
## 9  UNDER.medium / AxcQ1 Homog.  0.53089374 -0.85525175      less    0.2              0.45  
## 10         ELEV / AxcQ2      r  0.06836369  0.75587260 two-sided  0.476 0.646714285714286  
## 11 DIST_TO_EDGE / AxcQ2      r -0.02305324 -0.64468775 two-sided  0.526 0.676285714285714  
## 12       CANOPY / AxcQ2      r -0.08514258 -1.44000659 two-sided  0.168             0.432  
## 13          TEN / AxcQ2      r -0.07075257 -1.66212636 two-sided  0.092            0.3744  
## 14        FIFTY / AxcQ2      r -0.03532906 -1.01730191 two-sided  0.291 0.523636363636364  
## 15         CECR / AxcQ2      r  0.05215427  2.00077290 two-sided   0.03            0.3744  
## 16  UNDER.dense / AxcQ2 Homog.  0.18278557  0.22867378      less  0.601           0.68625  
## 17    UNDER.low / AxcQ2 Homog.  0.23598540 -0.03940811      less  0.503 0.646714285714286  
## 18 UNDER.medium / AxcQ2 Homog.  0.57870228  0.70731698      less  0.796 0.842823529411765  
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Results can be represented using a table with colors indicating significance :

## [1] "juveniles"

## [1] "adults"

Significance with axes can also be reported on the factorial map of RLQ analysis. Here, significant associations with the first axis are represented in blue, with the second axis in orange, with both axes in green (variables with no significant association are in black)

## [1] "juveniles"

## [1] "adults"